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81:
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1952:
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401:
The concept of rigged
Hilbert space places this idea in an abstract functional-analytic framework. Formally, a rigged Hilbert space consists of a
2604:
1589:
2419:
1244:
262:
on
Hilbert space can be formulated. "Rigged Hilbert spaces are well known as the structure which provides a proper mathematical meaning to the
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35:
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567:
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169:
99:
49:
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902:
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1485:
1421:. Generalized Functions, vol. 4: Some Applications of Harmonic Analysis. Rigged Hilbert Spaces. Academic Press, New York, 1964.
2828:
2201:
2196:
2189:
2125:
2009:
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2434:
1945:
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1342:
381:. To properly consider this function as an eigenfunction requires some way of stepping outside the strict confines of the
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1692:
2681:
2060:
1919:
1641:
1626:
1454:
1218:
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479:
in its 'test function' topology, is realised as a space of distributions or generalised functions of some sort, and the
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323:
130:
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1337:
421:
277:
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2444:
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1901:
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140:
134:
126:
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1855:
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1089:
625:
445:
386:
224:
2040:
1896:
2551:
2414:
2247:
2232:
2004:
1712:
1156:
41:
2804:
2133:
1646:
151:
2838:
2733:
2143:
2014:
1938:
1748:
1549:
866:
798:{\displaystyle H=L^{2}(\mathbb {R} ^{n}),\ \Phi =H^{s}(\mathbb {R} ^{n}),\ \Phi ^{*}=H^{-s}(\mathbb {R} ^{n}),}
1621:
2833:
2506:
2481:
2299:
2288:
1999:
1845:
1438:
R. de la Madrid, "The role of the rigged
Hilbert space in Quantum Mechanics," Eur. J. Phys. 26, 287 (2005);
2357:
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1057:
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1138:. In the case of complex Hilbert spaces, we use a Hermitian inner product; it will be complex linear in
318:
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635:
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259:
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1418:
1401:
1388:
1360:
480:
368:
228:
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2110:
2065:
1989:
1835:
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1682:
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808:
372:
255:
200:
Construction linking the study of "bound" and continuous eigenvalues in functional analysis
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2536:
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2107:
2094:
2019:
1961:
1697:
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236:
1524:
1150:(physics convention), and conjugate-linear (complex anti-linear) in the other variable.
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2184:
2091:
2024:
1984:
1886:
1738:
1539:
1414:
1203:
415:
1426:
Generalized
Eigenfunction Expansions and Unitary Representations of Topological Groups
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1810:
1800:
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244:
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1387:, Arno Bohm, Heinz-Dietrich Doebner, Piotr Kielanowski, eds., Springer-Verlag,
1994:
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463:
1979:
1965:
453:
359:
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2511:
1439:
1431:
R. de la Madrid, "Quantum
Mechanics in Rigged Hilbert Space Language,"
1047:{\displaystyle \langle u,v\rangle _{\Phi \times \Phi ^{*}}=(u,v)_{H}}
1385:
Irreversibility and
Causality, Semigroups and Rigged Hilbert Spaces
2582:
2586:
1934:
1450:
537:
are faithfully represented as distributions (because we assume
111:
56:
15:
1202:
is often named the "Gelfand triple" (after the mathematician
616:; this comment is an abstract expression of the idea that
1305:{\displaystyle i^{*}i:\Phi \subset H=H^{*}\to \Phi ^{*}.}
1225:
is a
Hilbert space in its own right, this isomorphism is
77:
1355:
Krasnoholovets, Volodymyr; Columbus, Frank H. (2004).
1247:
1159:
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983:
905:
811:
667:
638:
570:
493:
424:
326:
280:
599:{\displaystyle \Phi \subseteq H\subseteq \Phi ^{*}.}
518:{\displaystyle \phi \mapsto \langle v,\phi \rangle }
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2402:
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2335:
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2124:
2079:
2033:
1972:
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1793:
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1731:
1670:
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1558:
1493:
462:for the Hilbert norm. We consider the inclusion of
72:
may be too technical for most readers to understand
2542:Spectral theory of ordinary differential equations
1806:Spectral theory of ordinary differential equations
1304:
1194:
1130:
1078:
1046:
953:
823:
797:
653:
632:: Here (in the simplest case of Sobolev spaces on
608:The most significant examples are those for which
598:
517:
436:
350:
305:
2440:SchröderâBernstein theorems for operator algebras
139:but its sources remain unclear because it lacks
418:, that is one for which the natural inclusion
393:theory was developed in the years after 1950.
385:theory. This was supplied by the apparatus of
2598:
1946:
1462:
1428:, Polish Scientific Publishers, Warsaw, 1968.
1229:the same as the composition of the inclusion
1131:{\displaystyle v\in H=H^{*}\subset \Phi ^{*}}
971:is then compatible with the inner product on
8:
997:
984:
512:
500:
1195:{\displaystyle (\Phi ,\,\,H,\,\,\Phi ^{*})}
954:{\displaystyle i^{*}:H=H^{*}\to \Phi ^{*}.}
50:Learn how and when to remove these messages
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223:) is a construction designed to link the
188:Learn how and when to remove this message
170:Learn how and when to remove this message
100:Learn how and when to remove this message
84:, without removing the technical details.
1759:Group algebra of a locally compact group
1359:. Nova Science Publishers. p. 79.
1320:
235:. Such spaces were introduced to study
1381:Quantum Mechanics Beyond Hilbert Space
264:Dirac formulation of quantum mechanics
2273:Spectral theory of normal C*-algebras
2071:Spectral theory of normal C*-algebras
628:. Also, a simple example is given by
82:make it understandable to non-experts
7:
2268:Spectral theory of compact operators
254:Using this notion, a version of the
1079:{\displaystyle u\in \Phi \subset H}
2420:CohenâHewitt factorization theorem
1290:
1264:
1180:
1163:
1119:
1067:
1008:
1001:
939:
835:Formal definition (Gelfand triple)
749:
708:
584:
571:
425:
14:
2729:Compact operator on Hilbert space
2425:Extensions of symmetric operators
351:{\displaystyle -i{\frac {d}{dx}}}
31:This article has multiple issues.
2243:Positive operator-valued measure
1915:
1914:
1841:Topological quantum field theory
654:{\displaystyle \mathbb {R} ^{n}}
437:{\displaystyle \Phi \subseteq H}
306:{\displaystyle x\mapsto e^{ix},}
116:
61:
20:
2527:RayleighâFaberâKrahn inequality
1410:(See paragraphs 23.8 and 23.32)
1357:New Research in Quantum Physics
620:consists of test functions and
560:. Therefore, the definition of
39:or discuss these issues on the
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1189:
1160:
1035:
1022:
935:
789:
774:
739:
724:
699:
684:
497:
284:
1:
2435:Limiting absorption principle
1637:Uniform boundedness principle
1397:(Provides a survey overview.)
2061:Singular value decomposition
963:The duality pairing between
861:a dense subspace, such that
546:Riesz representation theorem
397:Functional analysis approach
2492:Hearing the shape of a drum
2175:Decomposition of a spectrum
1338:Encyclopedia of Mathematics
564:is in terms of a sandwich:
410:, together with a subspace
239:. They bring together the '
2862:
2698:Hilbert projection theorem
2080:Special Elements/Operators
1780:Invariant subspace problem
2677:CauchyâSchwarz inequality
2552:Superstrong approximation
2415:Banach algebra cohomology
2248:Projection-valued measure
2233:Borel functional calculus
2005:Projection-valued measure
1910:
1500:
391:generalized eigenfunction
2144:Spectrum of a C*-algebra
2015:Spectrum of a C*-algebra
1749:Spectrum of a C*-algebra
869:structure for which the
867:topological vector space
125:This article includes a
2572:WienerâKhinchin theorem
2507:Kuznetsov trace formula
2482:Almost Mathieu operator
2300:Banach function algebra
2289:Amenable Banach algebra
2046:GelfandâNaimark theorem
2000:Noncommutative topology
1846:Noncommutative geometry
154:more precise citations.
2844:Schwartz distributions
2547:SturmâLiouville theory
2445:ShermanâTakeda theorem
2325:TomitaâTakesaki theory
2100:Hermitian/Self-adjoint
2051:Gelfand representation
1902:TomitaâTakesaki theory
1877:Approximation property
1821:Calculus of variations
1333:"Rigged_Hilbert_space"
1306:
1209:Note that even though
1196:
1132:
1080:
1048:
955:
825:
824:{\displaystyle s>0}
799:
655:
600:
519:
475:. The latter, dual to
444:is continuous. It is
438:
352:
307:
221:equipped Hilbert space
2829:Generalized functions
2708:Polarization identity
2651:Orthogonal complement
2041:GelfandâMazur theorem
1897:BanachâMazur distance
1860:Generalized functions
1383:(1996), appearing in
1307:
1221:) if it happens that
1197:
1144:(math convention) or
1133:
1081:
1049:
977:, in the sense that:
956:
826:
800:
656:
624:of the corresponding
601:
520:
439:
353:
319:differential operator
308:
2682:Riesz representation
2637:L-semi-inner product
2517:Proto-value function
2496:Dirichlet eigenvalue
2410:Abstract index group
2295:Approximate identity
2258:Rigged Hilbert space
2134:KreinâRutman theorem
1980:Involution/*-algebra
1642:Kakutani fixed-point
1627:Riesz representation
1245:
1219:Riesz representation
1157:
1090:
1058:
981:
903:
887:with its dual space
841:rigged Hilbert space
809:
665:
636:
568:
562:rigged Hilbert space
544:Now by applying the
491:
422:
324:
278:
217:nested Hilbert space
209:rigged Hilbert space
2703:Parseval's identity
2672:Bessel's inequality
2320:Von Neumann algebra
2056:Polar decomposition
1826:Functional calculus
1785:Mahler's conjecture
1764:Von Neumann algebra
1478:Functional analysis
274:A function such as
260:unbounded operators
249:continuous spectrum
233:functional analysis
2450:Unbounded operator
2379:Essential spectrum
2358:SchurâHorn theorem
2348:BauerâFike theorem
2343:AlonâBoppana bound
2336:Finite-Dimensional
2310:Nuclear C*-algebra
2154:Spectral asymmetry
1851:Riemann hypothesis
1550:Topological vector
1406:ĂlĂ©ments d'analyse
1302:
1192:
1128:
1076:
1044:
951:
821:
795:
651:
596:
515:
481:linear functionals
434:
348:
303:
127:list of references
2816:
2815:
2759:Sesquilinear form
2712:Parallelogram law
2656:Orthonormal basis
2580:
2579:
2557:Transfer operator
2532:Spectral geometry
2217:Spectral abscissa
2197:Approximate point
2139:Normal eigenvalue
1928:
1927:
1831:Integral operator
1608:
1607:
1366:978-1-59454-001-1
1235:with its adjoint
1213:is isomorphic to
893:, the adjoint to
857:a Hilbert space,
747:
707:
369:square-integrable
346:
251:', in one place.
229:square-integrable
198:
197:
190:
180:
179:
172:
110:
109:
102:
54:
2851:
2646:Prehilbert space
2607:
2600:
2593:
2584:
2562:Transform theory
2282:Special algebras
2263:Spectral theorem
2226:Spectral Theorem
2066:Spectral theorem
1955:
1948:
1941:
1932:
1918:
1917:
1836:Jones polynomial
1754:Operator algebra
1498:
1471:
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1457:
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1440:quant-ph/0502053
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650:
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605:
603:
602:
597:
592:
591:
559:
553:
548:we can identify
540:
536:
530:
524:
522:
521:
516:
486:
483:on the subspace
478:
474:
470:
461:
451:
443:
441:
440:
435:
414:which carries a
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409:
380:
366:
357:
355:
354:
349:
347:
345:
334:
312:
310:
309:
304:
299:
298:
256:spectral theorem
193:
186:
175:
168:
164:
161:
155:
150:this article by
141:inline citations
120:
119:
112:
105:
98:
94:
91:
85:
65:
64:
57:
46:
24:
23:
16:
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2849:
2848:
2839:Spectral theory
2819:
2818:
2817:
2812:
2805:SegalâBargmann
2773:
2744:HilbertâSchmidt
2734:Densely defined
2717:
2686:
2660:
2616:
2611:
2581:
2576:
2537:Spectral method
2522:Ramanujan graph
2470:
2454:
2430:Fredholm theory
2398:
2393:Shilov boundary
2389:Structure space
2367:Generalizations
2362:
2353:Numerical range
2331:
2315:Uniform algebra
2277:
2253:Riesz projector
2238:Min-max theorem
2221:
2207:Direct integral
2163:
2149:Spectral radius
2120:
2075:
2029:
2020:Spectral radius
1968:
1962:Spectral theory
1959:
1929:
1924:
1906:
1870:Advanced topics
1865:
1789:
1768:
1727:
1693:HilbertâSchmidt
1666:
1657:GelfandâNaimark
1604:
1554:
1489:
1475:
1445:
1419:N. Ya. Vilenkin
1379:J.-P. Antoine,
1375:
1374:
1367:
1354:
1353:
1349:
1327:
1326:
1322:
1317:
1289:
1276:
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1155:
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1139:
1118:
1105:
1088:
1087:
1056:
1055:
1034:
1007:
996:
979:
978:
972:
968:
964:
938:
925:
906:
901:
900:
894:
888:
882:
878:is continuous.
873:
862:
858:
852:
844:
837:
807:
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777:
761:
748:
727:
714:
687:
674:
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565:
555:
549:
538:
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526:
489:
488:
484:
476:
472:
466:
457:
449:
448:to assume that
420:
419:
411:
405:
399:
376:
371:for the usual (
362:
338:
322:
321:
287:
276:
275:
272:
237:spectral theory
201:
194:
183:
182:
181:
176:
165:
159:
156:
145:
131:related reading
121:
117:
106:
95:
89:
86:
78:help improve it
75:
66:
62:
25:
21:
12:
11:
5:
2859:
2858:
2855:
2847:
2846:
2841:
2836:
2834:Hilbert spaces
2831:
2821:
2820:
2814:
2813:
2811:
2810:
2802:
2796:compact &
2781:
2779:
2775:
2774:
2772:
2771:
2766:
2761:
2756:
2751:
2746:
2741:
2739:Hermitian form
2736:
2731:
2725:
2723:
2719:
2718:
2716:
2715:
2705:
2700:
2694:
2692:
2688:
2687:
2685:
2684:
2679:
2674:
2668:
2666:
2662:
2661:
2659:
2658:
2653:
2648:
2639:
2630:
2624:
2622:
2621:Basic concepts
2618:
2617:
2614:Hilbert spaces
2612:
2610:
2609:
2602:
2595:
2587:
2578:
2577:
2575:
2574:
2569:
2564:
2559:
2554:
2549:
2544:
2539:
2534:
2529:
2524:
2519:
2514:
2509:
2504:
2499:
2489:
2487:Corona theorem
2484:
2478:
2476:
2472:
2471:
2469:
2468:
2466:Wiener algebra
2462:
2460:
2456:
2455:
2453:
2452:
2447:
2442:
2437:
2432:
2427:
2422:
2417:
2412:
2406:
2404:
2400:
2399:
2397:
2396:
2386:
2384:Pseudospectrum
2381:
2376:
2374:Dirac spectrum
2370:
2368:
2364:
2363:
2361:
2360:
2355:
2350:
2345:
2339:
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2333:
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2329:
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2312:
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2302:
2297:
2291:
2285:
2283:
2279:
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2255:
2250:
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2240:
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2227:
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2222:
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2214:
2209:
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2192:
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2165:
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2113:
2105:
2097:
2089:
2083:
2081:
2077:
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2074:
2073:
2068:
2063:
2058:
2053:
2048:
2043:
2037:
2035:
2031:
2030:
2028:
2027:
2025:Operator space
2022:
2017:
2012:
2007:
2002:
1997:
1992:
1987:
1985:Banach algebra
1982:
1976:
1974:
1973:Basic concepts
1970:
1969:
1960:
1958:
1957:
1950:
1943:
1935:
1926:
1925:
1923:
1922:
1911:
1908:
1907:
1905:
1904:
1899:
1894:
1889:
1887:Choquet theory
1884:
1879:
1873:
1871:
1867:
1866:
1864:
1863:
1853:
1848:
1843:
1838:
1833:
1828:
1823:
1818:
1813:
1808:
1803:
1797:
1795:
1791:
1790:
1788:
1787:
1782:
1776:
1774:
1770:
1769:
1767:
1766:
1761:
1756:
1751:
1746:
1741:
1739:Banach algebra
1735:
1733:
1729:
1728:
1726:
1725:
1720:
1715:
1710:
1705:
1700:
1695:
1690:
1685:
1680:
1674:
1672:
1668:
1667:
1665:
1664:
1662:BanachâAlaoglu
1659:
1654:
1649:
1644:
1639:
1634:
1629:
1624:
1618:
1616:
1610:
1609:
1606:
1605:
1603:
1602:
1597:
1592:
1590:Locally convex
1587:
1573:
1568:
1562:
1560:
1556:
1555:
1553:
1552:
1547:
1542:
1537:
1532:
1527:
1522:
1517:
1512:
1507:
1501:
1495:
1491:
1490:
1476:
1474:
1473:
1466:
1459:
1451:
1444:
1443:
1436:
1429:
1422:
1412:
1399:
1376:
1373:
1372:
1365:
1347:
1319:
1318:
1316:
1313:
1301:
1296:
1292:
1288:
1283:
1279:
1275:
1272:
1269:
1266:
1263:
1260:
1255:
1251:
1204:Israel Gelfand
1191:
1186:
1182:
1176:
1173:
1168:
1165:
1162:
1125:
1121:
1117:
1112:
1108:
1104:
1101:
1098:
1095:
1075:
1072:
1069:
1066:
1063:
1041:
1037:
1033:
1030:
1027:
1024:
1021:
1014:
1010:
1006:
1003:
999:
995:
992:
989:
986:
950:
945:
941:
937:
932:
928:
924:
921:
918:
913:
909:
836:
833:
820:
817:
814:
794:
791:
786:
781:
776:
771:
768:
764:
760:
755:
751:
744:
741:
736:
731:
726:
721:
717:
713:
710:
704:
701:
696:
691:
686:
681:
677:
673:
670:
648:
643:
630:Sobolev spaces
595:
590:
586:
582:
579:
576:
573:
514:
511:
508:
505:
502:
499:
496:
433:
430:
427:
416:finer topology
398:
395:
344:
341:
337:
332:
329:
302:
297:
294:
290:
286:
283:
271:
268:
213:Gelfand triple
199:
196:
195:
178:
177:
135:external links
124:
122:
115:
108:
107:
69:
67:
60:
55:
29:
28:
26:
19:
13:
10:
9:
6:
4:
3:
2:
2857:
2856:
2845:
2842:
2840:
2837:
2835:
2832:
2830:
2827:
2826:
2824:
2809:
2808:
2803:
2801:
2799:
2795:
2791:
2787:
2783:
2782:
2780:
2776:
2770:
2767:
2765:
2762:
2760:
2757:
2755:
2752:
2750:
2747:
2745:
2742:
2740:
2737:
2735:
2732:
2730:
2727:
2726:
2724:
2720:
2713:
2709:
2706:
2704:
2701:
2699:
2696:
2695:
2693:
2691:Other results
2689:
2683:
2680:
2678:
2675:
2673:
2670:
2669:
2667:
2663:
2657:
2654:
2652:
2649:
2647:
2643:
2642:Hilbert space
2640:
2638:
2634:
2633:Inner product
2631:
2629:
2626:
2625:
2623:
2619:
2615:
2608:
2603:
2601:
2596:
2594:
2589:
2588:
2585:
2573:
2570:
2568:
2565:
2563:
2560:
2558:
2555:
2553:
2550:
2548:
2545:
2543:
2540:
2538:
2535:
2533:
2530:
2528:
2525:
2523:
2520:
2518:
2515:
2513:
2510:
2508:
2505:
2503:
2500:
2497:
2493:
2490:
2488:
2485:
2483:
2480:
2479:
2477:
2473:
2467:
2464:
2463:
2461:
2457:
2451:
2448:
2446:
2443:
2441:
2438:
2436:
2433:
2431:
2428:
2426:
2423:
2421:
2418:
2416:
2413:
2411:
2408:
2407:
2405:
2403:Miscellaneous
2401:
2394:
2390:
2387:
2385:
2382:
2380:
2377:
2375:
2372:
2371:
2369:
2365:
2359:
2356:
2354:
2351:
2349:
2346:
2344:
2341:
2340:
2338:
2334:
2326:
2323:
2322:
2321:
2318:
2316:
2313:
2311:
2308:
2306:
2303:
2301:
2298:
2296:
2292:
2290:
2287:
2286:
2284:
2280:
2274:
2271:
2269:
2266:
2264:
2261:
2259:
2256:
2254:
2251:
2249:
2246:
2244:
2241:
2239:
2236:
2234:
2231:
2230:
2228:
2224:
2218:
2215:
2213:
2210:
2208:
2205:
2203:
2200:
2198:
2195:
2191:
2188:
2186:
2183:
2181:
2178:
2177:
2176:
2173:
2172:
2170:
2168:Decomposition
2166:
2160:
2157:
2155:
2152:
2150:
2147:
2145:
2142:
2140:
2137:
2135:
2132:
2131:
2129:
2127:
2123:
2117:
2114:
2112:
2109:
2106:
2104:
2101:
2098:
2096:
2093:
2090:
2088:
2085:
2084:
2082:
2078:
2072:
2069:
2067:
2064:
2062:
2059:
2057:
2054:
2052:
2049:
2047:
2044:
2042:
2039:
2038:
2036:
2032:
2026:
2023:
2021:
2018:
2016:
2013:
2011:
2008:
2006:
2003:
2001:
1998:
1996:
1993:
1991:
1988:
1986:
1983:
1981:
1978:
1977:
1975:
1971:
1967:
1963:
1956:
1951:
1949:
1944:
1942:
1937:
1936:
1933:
1921:
1913:
1912:
1909:
1903:
1900:
1898:
1895:
1893:
1892:Weak topology
1890:
1888:
1885:
1883:
1880:
1878:
1875:
1874:
1872:
1868:
1861:
1857:
1854:
1852:
1849:
1847:
1844:
1842:
1839:
1837:
1834:
1832:
1829:
1827:
1824:
1822:
1819:
1817:
1816:Index theorem
1814:
1812:
1809:
1807:
1804:
1802:
1799:
1798:
1796:
1792:
1786:
1783:
1781:
1778:
1777:
1775:
1773:Open problems
1771:
1765:
1762:
1760:
1757:
1755:
1752:
1750:
1747:
1745:
1742:
1740:
1737:
1736:
1734:
1730:
1724:
1721:
1719:
1716:
1714:
1711:
1709:
1706:
1704:
1701:
1699:
1696:
1694:
1691:
1689:
1686:
1684:
1681:
1679:
1676:
1675:
1673:
1669:
1663:
1660:
1658:
1655:
1653:
1650:
1648:
1645:
1643:
1640:
1638:
1635:
1633:
1630:
1628:
1625:
1623:
1620:
1619:
1617:
1615:
1611:
1601:
1598:
1596:
1593:
1591:
1588:
1585:
1581:
1577:
1574:
1572:
1569:
1567:
1564:
1563:
1561:
1557:
1551:
1548:
1546:
1543:
1541:
1538:
1536:
1533:
1531:
1528:
1526:
1523:
1521:
1518:
1516:
1513:
1511:
1508:
1506:
1503:
1502:
1499:
1496:
1492:
1487:
1483:
1479:
1472:
1467:
1465:
1460:
1458:
1453:
1452:
1449:
1441:
1437:
1434:
1430:
1427:
1423:
1420:
1416:
1415:I. M. Gelfand
1413:
1411:
1407:
1403:
1400:
1398:
1394:
1393:3-540-64305-2
1390:
1386:
1382:
1378:
1377:
1368:
1362:
1358:
1351:
1348:
1344:
1340:
1339:
1334:
1330:
1329:Minlos, R. A.
1324:
1321:
1314:
1312:
1299:
1294:
1281:
1277:
1273:
1270:
1267:
1261:
1258:
1253:
1249:
1239:
1233:
1228:
1220:
1207:
1205:
1184:
1174:
1171:
1166:
1151:
1148:
1142:
1123:
1115:
1110:
1106:
1102:
1099:
1096:
1093:
1073:
1070:
1064:
1061:
1039:
1031:
1028:
1025:
1019:
1012:
1004:
993:
990:
987:
975:
961:
948:
943:
930:
926:
922:
919:
916:
911:
907:
897:
891:
885:
879:
876:
872:
871:inclusion map
868:
855:
848:
842:
834:
832:
818:
815:
812:
792:
784:
769:
766:
762:
758:
753:
742:
734:
719:
715:
711:
702:
694:
679:
675:
671:
668:
646:
631:
627:
626:distributions
615:
614:nuclear space
606:
593:
588:
580:
577:
574:
563:
558:
552:
547:
542:
535:
529:
509:
506:
503:
494:
482:
469:
465:
460:
455:
447:
431:
428:
417:
408:
404:
403:Hilbert space
396:
394:
392:
388:
387:distributions
384:
383:Hilbert space
379:
375:) measure on
374:
370:
365:
361:
342:
339:
335:
330:
327:
320:
316:
315:eigenfunction
300:
295:
292:
288:
281:
269:
267:
265:
261:
257:
252:
250:
246:
242:
238:
234:
230:
226:
222:
218:
214:
210:
206:
192:
189:
174:
171:
163:
153:
149:
143:
142:
136:
132:
128:
123:
114:
113:
104:
101:
93:
83:
79:
73:
70:This article
68:
59:
58:
53:
51:
44:
43:
38:
37:
32:
27:
18:
17:
2806:
2797:
2793:
2789:
2785:
2754:Self-adjoint
2665:Main results
2475:Applications
2305:Disk algebra
2257:
2159:Spectral gap
2034:Main results
1882:Balanced set
1856:Distribution
1794:Applications
1647:KreinâMilman
1632:Closed graph
1425:
1409:
1408:VII (1978).
1405:
1402:J. Dieudonné
1396:
1384:
1380:
1356:
1350:
1336:
1323:
1237:
1231:
1226:
1208:
1152:
1146:
1140:
973:
962:
895:
889:
883:
881:Identifying
880:
874:
853:
846:
840:
838:
607:
561:
556:
550:
543:
533:
527:
467:
458:
406:
400:
390:
377:
367:, but isn't
363:
273:
253:
225:distribution
220:
216:
212:
208:
202:
184:
166:
157:
146:Please help
138:
96:
87:
71:
47:
40:
34:
33:Please help
30:
2764:Trace class
2502:Heat kernel
2202:Compression
2087:Isospectral
1811:Heat kernel
1801:Hardy space
1708:Trace class
1622:HahnâBanach
1584:Topological
1424:K. Maurin,
1153:The triple
899:is the map
865:is given a
464:dual spaces
245:eigenvector
241:bound state
231:aspects of
205:mathematics
152:introducing
2823:Categories
2180:Continuous
1995:C*-algebra
1990:B*-algebra
1744:C*-algebra
1559:Properties
1433:PhD Thesis
1315:References
843:is a pair
270:Motivation
36:improve it
1966:-algebras
1718:Unbounded
1713:Transpose
1671:Operators
1600:Separable
1595:Reflexive
1580:Algebraic
1566:Barrelled
1343:EMS Press
1331:(2001) ,
1295:∗
1291:Φ
1287:→
1282:∗
1268:⊂
1265:Φ
1254:∗
1185:∗
1181:Φ
1164:Φ
1124:∗
1120:Φ
1116:⊂
1111:∗
1097:∈
1071:⊂
1068:Φ
1065:∈
1054:whenever
1013:∗
1009:Φ
1005:×
1002:Φ
998:⟩
985:⟨
944:∗
940:Φ
936:→
931:∗
912:∗
767:−
754:∗
750:Φ
709:Φ
589:∗
585:Φ
581:⊆
575:⊆
572:Φ
513:⟩
510:ϕ
501:⟨
498:↦
495:ϕ
429:⊆
426:Φ
360:real line
328:−
285:↦
160:July 2023
90:June 2020
42:talk page
2778:Examples
2567:Weyl law
2512:Lax pair
2459:Examples
2293:With an
2212:Discrete
2190:Residual
2126:Spectrum
2111:operator
2103:operator
2095:operator
2010:Spectrum
1920:Category
1732:Algebras
1614:Theorems
1571:Complete
1540:Schwartz
1486:glossary
541:dense).
487:of type
389:, and a
373:Lebesgue
2792:) with
2769:Unitary
2628:Adjoint
2108:Unitary
1723:Unitary
1703:Nuclear
1688:Compact
1683:Bounded
1678:Adjoint
1652:Minâmax
1545:Sobolev
1530:Nuclear
1520:Hilbert
1515:Fréchet
1480: (
1435:(2001).
446:no loss
358:on the
317:of the
247:) and '
148:improve
76:Please
2749:Normal
2092:Normal
1698:Normal
1535:Orlicz
1525:Hölder
1505:Banach
1494:Spaces
1482:topics
1391:
1363:
805:where
746:
706:
313:is an
2800:<â
2185:Point
1510:Besov
1217:(via
851:with
612:is a
554:with
454:dense
133:, or
2722:Maps
2644:and
2635:and
2116:Unit
1964:and
1858:(or
1576:Dual
1417:and
1389:ISBN
1361:ISBN
1086:and
967:and
849:, Ί)
816:>
525:for
258:for
227:and
207:, a
1227:not
1206:).
531:in
471:in
456:in
452:is
266:."
243:' (
203:In
80:to
2825::
1484:â
1404:,
1395:.
1341:,
1335:,
839:A
831:.
661:)
622:Ί*
219:,
215:,
137:,
129:,
45:.
2807:F
2798:n
2794:K
2790:K
2788:(
2786:C
2714:)
2710:(
2606:e
2599:t
2592:v
2498:)
2494:(
2395:)
2391:(
1954:e
1947:t
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1582:/
1578:(
1488:)
1470:e
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1442:.
1369:.
1300:.
1278:H
1274:=
1271:H
1262::
1259:i
1250:i
1240:*
1238:i
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1215:Ί
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1190:)
1175:,
1172:H
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1161:(
1147:v
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1103:=
1100:H
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1026:u
1023:(
1020:=
994:v
991:,
988:u
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969:Ί
965:Ί
949:.
927:H
923:=
920:H
917::
908:i
896:i
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884:H
875:i
863:Ί
859:Ί
854:H
847:H
845:(
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813:s
793:,
790:)
785:n
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775:(
770:s
763:H
759:=
743:,
740:)
735:n
730:R
725:(
720:s
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712:=
703:,
700:)
695:n
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685:(
680:2
676:L
672:=
669:H
647:n
642:R
618:Ί
610:Ί
594:.
578:H
557:H
551:H
539:Ί
534:H
528:v
507:,
504:v
485:Ί
477:Ί
473:Ί
468:H
459:H
450:Ί
432:H
412:Ί
407:H
378:R
364:R
343:x
340:d
336:d
331:i
301:,
296:x
293:i
289:e
282:x
211:(
191:)
185:(
173:)
167:(
162:)
158:(
144:.
103:)
97:(
92:)
88:(
74:.
52:)
48:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.